Henrik Aalborg Nielsen

Time-adaptive quantile regression

An algorithm for time-adaptive quantile regression is presented. The algorithm is based on the simplex algorithm, and the linear optimization formulation of the quantile regression problem is given. The observations have been split to allow a direct use of the simplex algorithm. The simplex method and an updating procedure are combined into a new algorithm for time-adaptive quantile regression, which generates new solutions on the basis of the old solution, leading to savings in computation time. The suggested algorithm is tested against a static quantile regression model on a data set with wind power production, where the models combine splines and quantile regression. The comparison indicates superior performance for the time-adaptive quantile regression in all the performance parameters considered.

Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations.

PurposeThe objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling.MethodsThe intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM® for parameter estimation in SDE models.ResultsVarious applications of SDEs in population PK/PD modeling are illustrated through a systematic model development example using clinical PK data of the gonadotropin releasing hormone (GnRH) antagonist degarelix. The dynamic noise estimates were used to track variations in model parameters and systematically build an absorption model for subcutaneously administered degarelix.ConclusionsThe EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained by tracking unexplained variations in parameters.

Forecasting Electricity Spot Prices Accounting for Wind Power Predictions

A two-step methodology for forecasting of electricity spot prices is introduced, with focus on the impact of predicted system load and wind power generation. The nonlinear and nonstationary influence of these explanatory variables is accommodated in a first step based on a nonparametric and time-varying regression model. In a second step, time-series models, i.e., ARMA and Holt-Winters, are applied to account for residual autocorrelation and seasonal dynamics. Empirical results are presented for out-of-sample forecasts of day-ahead prices in the Western Danish price area of Nord Pools Elspot, during a two year period covering 2010-2011. These results clearly demonstrate the practical benefits of accounting for the complex influence of these explanatory variables.

From wind ensembles to probabilistic information about future wind power production -- results from an actual application

Meteorological ensemble forecasts aim at quantifying the uncertainty of the future development of the weather by supplying several possible scenarios of this development. Here we address the use of such scenarios in probabilistic forecasting of wind power production. Specifically, for each forecast horizon we aim at supplying quantiles of the wind power production conditional on the information available at the time at which the forecast is generated. This involves: (i) transformation of meteorological ensemble forecasts into wind power ensemble forecasts and (ii) calculation of quantiles based on the wind power ensemble forecasts. Given measurements of power production, representing a region or a single wind farm, we have developed methods applicable for these two steps. While (ii) should in principle be a simple task we found that the probabilistic information contained in the wind power ensembles from (i) cannot be used directly and therefore both (i) and (ii) requires statistical modelling. Based on these findings an demo-application, supplying quantile forecasts for operational horizons of up to approximately 6 days, was developed for two utilities participating in a common project. The application use ECMWF-ensembles. One setup corresponds to an offshore wind farm (Nysted, Denmark) and one corresponds to regional forecasting (Western Denmark). In the paper we analyze the results obtained from 8 months of actual operation of this system. It is concluded that the demo-application produce reliable forecasts. The average difference between the 75% and 25% quantile forecasts exceeds 50% of the installed capacity for horizons longer than approximately 4 days for the wind farm setup. For the regional forecasts the corresponding horizon is not reached within 7 days, which is the maximum horizon available. The ability of the demo-application to differentiate between situations with low and high uncertainty is analysed. Also, the relation between the forecasted uncertainty and the actual skill of a point forecast is analysed. A satisfactory agreement is observed

Local linear regression with adaptive orthogonal fitting for the wind power application

Abstract Short-term forecasting of wind generation requires a model of the function for the conversion of meteorological variables (mainly wind speed) to power production. Such a power curve is nonlinear and bounded, in addition to being nonstationary. Local linear regression is an appealing nonparametric approach for power curve estimation, for which the model coefficients can be tracked with recursive Least Squares (LS) methods. This may lead to an inaccurate estimate of the true power curve, owing to the assumption that a noise component is present on the response variable axis only. Therefore, this assumption is relaxed here, by describing a local linear regression with orthogonal fit. Local linear coefficients are defined as those which minimize a weighted Total Least Squares (TLS) criterion. An adaptive estimation method is introduced in order to accommodate nonstationarity. This has the additional benefit of lowering the computational costs of updating local coefficients every time new observations become available. The estimation method is based on tracking the left-most eigenvector of the augmented covariance matrix. A robustification of the estimation method is also proposed. Simulations on semi-artificial datasets (for which the true power curve is available) underline the properties of the proposed regression and related estimation methods. An important result is the significantly higher ability of local polynomial regression with orthogonal fit to accurately approximate the target regression, even though it may hardly be visible when calculating error criteria against corrupted data.

A generalization of some classical time series tools

In classical time series analysis the sample autocorrelation function (SACF) and the sample partial autocorrelation function (SPACF) has gained wide application for structural identification of linear time series models. We suggest generalizations, founded on smoothing techniques, applicable for structural identification of non-linear time series models. A similar generalization of the sample cross correlation function is discussed. Furthermore, a measure of the departure from linearity is suggested. It is shown how bootstrapping can be applied to construct confidence intervals under independence or linearity. The generalizations do not prescribe a particular smoothing technique. In fact, when the smoother is replaced by a linear regression the generalizations reduce to close approximations of SACF and SPACF. For this reason a smooth transition from the linear to the non-linear case can be obtained by varying the bandwidth of a local linear smoother. By adjusting the flexibility of the smoother, the power of the tests for independence and linearity against specific alternatives can be adjusted. The generalizations allow for graphical presentations, very similar to those used for SACF and SPACF. In this paper the generalizations are applied to some simulated data sets and to the Canadian lynx data. The generalizations seem to perform well and the measure of the departure from linearity proves to be an important additional tool.

Non-linear mixed-effects pharmacokinetic/pharmacodynamic modelling in NLME using differential equations

The standard software for non-linear mixed-effect analysis of pharmacokinetic/pharmacodynamic (PK/PD) data is NONMEM while the non-linear mixed-effects package NLME is an alternative as long as the models are fairly simple. We present the nlmeODE package which combines the ordinary differential equation (ODE) solver package odesolve and the non-linear mixed effects package NLME thereby enabling the analysis of complicated systems of ODEs by non-linear mixed-effects modelling. The pharmacokinetics of the anti-asthmatic drug theophylline is used to illustrate the applicability of the nlmeODE package for population PK/PD analysis using the available data analysis tools in R for model inspection and validation. The nlmeODE package is numerically stable and provides accurate parameter estimates which are consistent with NONMEM estimates.

Pharmacokinetic/Pharmacodynamic Modelling of GnRH Antagonist Degarelix: A Comparison of the Non-linear Mixed-Effects Programs NONMEM and NLME

In this paper, the two non-linear mixed-effects programs NONMEM and NLME were compared for their use in population pharmacokinetic/pharmacodynamic (PK/PD) modelling. We have described the first-order conditional estimation (FOCE) method as implemented in NONMEM and the alternating algorithm in NLME proposed by Lindstrom and Bates. The two programs were tested using clinical PK/PD data of a new gonadotropin-releasing hormone (GnRH) antagonist degarelix currently being developed for prostate cancer treatment. The pharmacokinetics of intravenous administered degarelix was analysed using a three compartment model while the pharmacodynamics was analysed using a turnover model with a pool compartment. The results indicated that the two algorithms produce consistent parameter estimates. The bias and precision of the two algorithms were further investigated using a parametric bootstrap procedure which showed that NONMEM produced more accurate results than NLME together with the nlmeODE package for this specific study.

Online short-term heat load forecasting for single family houses

This paper presents a method for forecasting the load for heating in a single-family house. Both space and hot tap water heating are forecasted. The forecasting model is built using data from sixteen houses in Sønderborg, Denmark, combined with local climate measurements and weather forecasts. Every hour the hourly heat load for each house the following two days is forecasted. The forecast models are adaptive linear time-series models and the climate inputs used are: ambient temperature, global radiation, and wind speed. A computationally efficient recursive least squares scheme is used. The models are optimized to fit the level of optimal adaptivity and the thermal dynamical response of the building. Identification of a model, which is suitable for application to all the houses, is carried out. The results show that the forecasting errors mainly are related to: unpredictable high frequency variations in the heat load signal (predominant only for some houses), peaks presumably from showers, shifts in resident behavior, and uncertainty of the weather forecasts for longer horizons, especially for the solar radiation.

On On-line Systems for Short-term Forecasting for Energy Systems

The paper describes experiences with developing on-line computer systems for short-term forecasting of wind power production and heat consumption in district heating networks. The computer systems are briefly described and some general aspects regarding system modeling with the purpose of forecasting are discussed. One consequence of the approach used is that the stochastic properties of the forecast errors can not be inferred from the models generating the forecasts. With the purpose of using the stochastic properties as input to formal OR-models we discuss how these can be modeled.